Quasi-optimal complexity $hp$-FEM for Poisson on a rectangle

We show, in one dimension, that an $hp$-Finite Element Method ($hp$-FEM) discretisation can be solved in optimal complexity because the discretisation has a special sparsity structure that ensures that the reverse Cholesky factorisation -- Cholesky starting from the bottom right instead of the top left -- remains sparse. Moreover, computing and inverting the factorisation almost entirely trivially parallelises across the different elements. By incorporating this approach into an Alternating Direction Implicit (ADI) method \`a la Fortunato and Townsend (2020) we can solve, within a prescribed tolerance, an $hp$-FEM discretisation of the (screened) Poisson equation on a rectangle, in parallel, with quasi-optimal complexity: O(N² log N) operations where N is the maximal total degrees of freedom in each dimension. When combined with fast Legendre transforms we can also solve nonlinear time-evolution partial differential equations in a quasi-optimal complexity of O(N² log² N) operations, which we demonstrate on the (viscid) Burgers' equation.